Scores for a common standardized college aptitude test are normally distributed with a mean of 497 and a standard deviation of 104. Randomly selected men are given a Test Prepartion Course before taking this test. Assume, for sake of argument, that the test has no effect.

If 1 of the men is randomly selected, find the probability that his score is at least 588.9.
P(X > 588.9) = Round to 4 decimal places.

NOTE: Answers obtained using exact z-scores or z-scores rounded to 2 decimal places are accepted.

If 8 of the men are randomly selected, find the probability that their mean score is at least 588.9.
P(X\displaystyle \overline{{X}} > 588.9) = Round to 4 decimal places.

NOTE: Answers obtained using exact z-scores or z-scores rounded to 2 decimal places are accepted.

If the random sample of 8 men does result in a mean score of 588.9, is there strong evidence to support the claim that the course is actually effective?